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Theorem cnref1o 10302
Description: There is a natural one-to-one mapping from  ( RR  X.  RR ) to  CC, where we map  <. x ,  y
>. to  ( x  +  ( _i  x.  y ) ). In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 8697), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
Hypothesis
Ref Expression
cnref1o.1  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
Assertion
Ref Expression
cnref1o  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Distinct variable group:    x, y
Allowed substitution hints:    F( x, y)

Proof of Theorem cnref1o
StepHypRef Expression
1 cnref1o.1 . . . . 5  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
2 ovex 5803 . . . . 5  |-  ( x  +  ( _i  x.  y ) )  e. 
_V
31, 2fnmpt2i 6113 . . . 4  |-  F  Fn  ( RR  X.  RR )
4 1st2nd2 6079 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
54fveq2d 5448 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
6 df-ov 5781 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
75, 6syl6eqr 2306 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
8 xp1st 6069 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
9 xp2nd 6070 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
10 oveq1 5785 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  ( _i  x.  y ) )  =  ( ( 1st `  z
)  +  ( _i  x.  y ) ) )
11 oveq2 5786 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( _i  x.  y )  =  ( _i  x.  ( 2nd `  z ) ) )
1211oveq2d 5794 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z )  +  ( _i  x.  y
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
13 ovex 5803 . . . . . . . . 9  |-  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  _V
1410, 12, 1, 13ovmpt2 5903 . . . . . . . 8  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
) F ( 2nd `  z ) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
158, 9, 14syl2anc 645 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
167, 15eqtrd 2288 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
178recnd 8815 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  CC )
18 ax-icn 8750 . . . . . . . 8  |-  _i  e.  CC
199recnd 8815 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  CC )
20 ax-mulcl 8753 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  ( 2nd `  z )  e.  CC )  -> 
( _i  x.  ( 2nd `  z ) )  e.  CC )
2118, 19, 20sylancr 647 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( _i  x.  ( 2nd `  z
) )  e.  CC )
2217, 21addcld 8808 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  CC )
2316, 22eqeltrd 2330 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  e.  CC )
2423rgen 2581 . . . 4  |-  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC
25 ffnfv 5605 . . . 4  |-  ( F : ( RR  X.  RR ) --> CC  <->  ( F  Fn  ( RR  X.  RR )  /\  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC ) )
263, 24, 25mpbir2an 891 . . 3  |-  F :
( RR  X.  RR )
--> CC
278, 9jca 520 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z )  e.  RR ) )
28 xp1st 6069 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 1st `  w )  e.  RR )
29 xp2nd 6070 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 2nd `  w )  e.  RR )
3028, 29jca 520 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( ( 1st `  w )  e.  RR  /\  ( 2nd `  w )  e.  RR ) )
31 cru 9692 . . . . . . 7  |-  ( ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  /\  ( ( 1st `  w
)  e.  RR  /\  ( 2nd `  w )  e.  RR ) )  ->  ( ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) )  <->  ( ( 1st `  z )  =  ( 1st `  w
)  /\  ( 2nd `  z )  =  ( 2nd `  w ) ) ) )
3227, 30, 31syl2an 465 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  =  ( ( 1st `  w )  +  ( _i  x.  ( 2nd `  w ) ) )  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
33 fveq2 5444 . . . . . . . . 9  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
34 fveq2 5444 . . . . . . . . . 10  |-  ( z  =  w  ->  ( 1st `  z )  =  ( 1st `  w
) )
35 fveq2 5444 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
3635oveq2d 5794 . . . . . . . . . 10  |-  ( z  =  w  ->  (
_i  x.  ( 2nd `  z ) )  =  ( _i  x.  ( 2nd `  w ) ) )
3734, 36oveq12d 5796 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
3833, 37eqeq12d 2270 . . . . . . . 8  |-  ( z  =  w  ->  (
( F `  z
)  =  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  <->  ( F `  w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
3938, 16vtoclga 2817 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( F `
 w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4016, 39eqeqan12d 2271 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
41 1st2nd2 6079 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  w  = 
<. ( 1st `  w
) ,  ( 2nd `  w ) >. )
424, 41eqeqan12d 2271 . . . . . . 7  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
43 fvex 5458 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
44 fvex 5458 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
4543, 44opth 4203 . . . . . . 7  |-  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. 
<->  ( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) )
4642, 45syl6bb 254 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
4732, 40, 463bitr4d 278 . . . . 5  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
4847biimpd 200 . . . 4  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
4948rgen2a 2582 . . 3  |-  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
50 dff13 5703 . . 3  |-  ( F : ( RR  X.  RR ) -1-1-> CC  <->  ( F :
( RR  X.  RR )
--> CC  /\  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
5126, 49, 50mpbir2an 891 . 2  |-  F :
( RR  X.  RR ) -1-1-> CC
52 ax-cnre 8764 . . . . . 6  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
53 oveq1 5785 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  y
) ) )
54 oveq2 5786 . . . . . . . . . 10  |-  ( y  =  v  ->  (
_i  x.  y )  =  ( _i  x.  v ) )
5554oveq2d 5794 . . . . . . . . 9  |-  ( y  =  v  ->  (
u  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  v
) ) )
56 ovex 5803 . . . . . . . . 9  |-  ( u  +  ( _i  x.  v ) )  e. 
_V
5753, 55, 1, 56ovmpt2 5903 . . . . . . . 8  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u F v )  =  ( u  +  ( _i  x.  v ) ) )
5857eqeq2d 2267 . . . . . . 7  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( w  =  ( u F v )  <-> 
w  =  ( u  +  ( _i  x.  v ) ) ) )
59582rexbiia 2550 . . . . . 6  |-  ( E. u  e.  RR  E. v  e.  RR  w  =  ( u F v )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
6052, 59sylibr 205 . . . . 5  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
61 fveq2 5444 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( F `  <. u ,  v >. )
)
62 df-ov 5781 . . . . . . . 8  |-  ( u F v )  =  ( F `  <. u ,  v >. )
6361, 62syl6eqr 2306 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( u F v ) )
6463eqeq2d 2267 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( w  =  ( F `  z
)  <->  w  =  (
u F v ) ) )
6564rexxp 4802 . . . . 5  |-  ( E. z  e.  ( RR 
X.  RR ) w  =  ( F `  z )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
6660, 65sylibr 205 . . . 4  |-  ( w  e.  CC  ->  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
)
6766rgen 2581 . . 3  |-  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
68 dffo3 5595 . . 3  |-  ( F : ( RR  X.  RR ) -onto-> CC  <->  ( F :
( RR  X.  RR )
--> CC  /\  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
) )
6926, 67, 68mpbir2an 891 . 2  |-  F :
( RR  X.  RR ) -onto-> CC
70 df-f1o 4674 . 2  |-  ( F : ( RR  X.  RR ) -1-1-onto-> CC  <->  ( F :
( RR  X.  RR ) -1-1-> CC  /\  F :
( RR  X.  RR ) -onto-> CC ) )
7151, 69, 70mpbir2an 891 1  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   <.cop 3603    X. cxp 4645    Fn wfn 4654   -->wf 4655   -1-1->wf1 4656   -onto->wfo 4657   -1-1-onto->wf1o 4658   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   1stc1st 6040   2ndc2nd 6041   CCcc 8689   RRcr 8690   _ici 8693    + caddc 8694    x. cmul 8696
This theorem is referenced by:  cnexALT  10303  cnrecnv  11601  cpnnen  12455  cnheiborlem  18400  mbfimaopnlem  18958
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378
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